21-355 Principles of Real Analysis I

9 units, Carnegie Mellon University

This course provides a rigorous and proof-based treatment of functions of one real variable. The course presumes some mathematical sophistication including the ability to recognize, read, and write proofs. Topics include: The Real Number System: Field and order axioms, sups and infs, completeness. Real Sequences. Bolzano-Weierstrass theorem. Topology of the Real Line: Open sets, closed sets, compactness, Heine-Borel Theorem. Continuity: extreme and intermediate value theorems, uniform continuity. Differentiation: chain rule, local extrema, mean-value theorem, L'Hospital's rule, Taylor's theorem. Riemann integration: sufficient conditions for integrability, fundamental theorems of calculus. (Three 50 minute lectures)

Prerequisites: (21-127 or 21-128 or 15-151) and 21-122

Offered: Spring 2027, Fall 2026, Summer 2026, Spring 2026, Fall 2025

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